TY  - JOUR
ID  - Shen2026
LB  - Shen2026
AU  - Shen, Shupei
AU  - He, Xiaoming
TI  - Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain
ST  - Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 4
SP  - 541
EP  - 562
DO  - https://doi.org/10.7494/OpMath.202606191
UR  - https://doi.org/10.7494/OpMath.202606191
AB  - In this paper, we study the existence of the solutions for the following \(p\)-Kirchhoff equation on bounded domain \[\begin{cases}-\Bigg(a+b\displaystyle\int_{\Omega}|\nabla u|^pdx\Bigg)\Delta_pu+\lambda|u|^{p-2}u=h(u), &x \in \Omega,\\ u=0, &x\in\partial\Omega,\end{cases}\] with prescribed mass \[\int_{\Omega}|u|^pdx=m^p,\] where \(2\leq p\lt 3\), \(a\gt 0\), \(b\gt 0\) are positive constants, \(h(u)\) is a general nonlinearity with Sobolev subcritical growth, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subset\mathbb{R}^3\) is a bounded domain, \(\lambda\in \mathbb{R}\) appears as a Lagrange multiplier. First, we prove that the equation admits a positive solution which is a local minimizer when \(h\) is \(L^p\)-subcritical or \(L^p\)-critical at infinity by Brezis-Nirenberg technique. Moreover, we get the multiplicity result by using the genus theory. Next, we prove that the equation has a local minimizer when \(h\) is \(L^p\)-supercritical. Besides, by using the Pohozaev analysis, we prove that in this case, if \(\Omega\) is a star-shaped domain with respect to the origin, the equation admits a second solution which is mountain pass type. To the best of our knowledge, this work seems to be the first contribution on the existence of normalized solutions for \(p\)-Kirchhoff equation on bounded domain.
KW  - \(p\)-Laplacian Kirchhoff equation
KW  - normalized solutions
KW  - existence and multiplicity
KW  - bounded domain
SN  - 1232-9274
ER  -